Summary and Analysis

Algebra I: Variation

Terms

Direct Variation, page 2

page 1 of 2

Direct Variation

The statement "
y
varies directly as
x
," means that when
x
increases,
y
increases by the same factor. In other words,
y
and
x
always have the same ratio:

= k

where
k
is the constant of variation. We can also express the relationship between
x
and
y
as:

y = kx

where
k
is the constant of variation.

Since
k
is constant (the same for every point), we can find
k
when given any point by dividing the y-coordinate by the x-coordinate. For example, if
y
varies directly as
x
, and
y = 6
when
x = 2
, the constant of variation is
k = = 3
. Thus, the equation describing this direct variation is
y = 3x
.

Example 1: If
y
varies directly as
x
, and
x = 12
when
y = 9
, what is the equation that describes this direct variation?

k = = y = x

Example 2: If
y
varies directly as
x
, and the constant of variation is
k =
, what is
y
when
x = 9
?

y = x = (9) = 15

As previously stated,
k
is constant for every point; i.e., the ratio between the
y
-coordinate of a point and the
x
-coordinate of a point is constant. Thus, given any two points
(x1, y1)
and
(x2, y2)
that satisfy the equation,
= k
and
= k
. Consequently,
=
for any two points that satisfy the equation.